Operational amplifiers

Building blocks of servos

Operational amplifiers (Op-amps)Want perfect amplifier• Infinite gain• Infinite input impedance

– will not load down source• Zero output impedance

– will drive anything

Have operational amplifiers• Gain ~ 106

• Input impedance ~ 100 M • Output impedance ~ 100 Problems• Gain too high

– slightest input noise causes max output• Other problems to be discussed laterSolutions• Use feedback• Gain depends only on resistance: Rf / Rin

– can control precisely

Op amp with feedback

V1 = (Vout - Vin) Rin/(Rin+Rf) + Vin

Vout = -A V1

Vout = -Vin (Rf / Rin) / [1 + (1+ Rf / Rin)/A]Small gain: Vout = -A Vin Rf / (Rf + Rin) ~ dividerLarge gain: Vout = -Vin (Rf / Rin) Note: V1 = -Vout / A ~ 0

Rin

-

+

Op amp

Vin Vout

RfSummingjunction

inverting

non-inverting

V1 If

Iin

Differential amplifier• Op amp output actually depends on voltage difference at two inputs

• Vout = - (Vin1 - Vin2) (Rf / Rin)

• Insensitivity to common voltage at both inputs = CMRR• Real op amps have problems with unbalanced input impedance

Solution: • Add input resistor and pot.• Op amp with built-in resistors = instrumentation amp.

Rin

-

+

Vin1 Vout

Rf

RinVin2

Non-ideal op-amp

~ Rf

Resistors to compensate for non-ideal op-amp

Ideal op-amp circuit

Rin

-

+

Vin1 Vout

Rf

inverting

non-inverting

V1

Vin2

Integrators • Put capacitor in op amp feedback path• Vout = - Vin (Zf / Rin) = - Vin / (2 jf C Rin)• Similar to low pass filter in high frequency limit

– except applies to low frequencies also– can show large gain near dc

• Recall V1 ~ 0 forces Iin = -Ifeedback

– charge on capacitor is integral of If

– since Vout = Q/Cf, Vout is integral of Iin

• Result is integrator– integration speed ~ 1 / RinCf

Integrator

Rin

-

+

Vin Vout

Cf

V1

Iin

If

log(Vout/Vin)

log(f )

Unity gain atf = 1 / 2 RC

Gain response

Phaseshift

log(f )Phase response

-90 degrees

0 degrees

Single-pole rolloff6 dB/octave= 10 dB/decade RC

0

>60dB

Shunted integrator• Limit dc gain• Advantages:

– dc input voltage no longer saturates op amp output– prevents servo runaway

• Dis-advantages– long term errors not well corrected by servo

log(Vout/Vin)

log(f )

Unity gain atf = 1 / 2 RinCf

Gain response

Phaseshift

log(f )Phase response

-90 degrees

0 degrees

Max gain = Rf/Rin

at f < 1 / 2 RfCf

0

Shunted integrator

Rin

-

+

Vin Vout

Cf

V1

Iin

If

Rf

Real op ampOp amp without feedback • Acts like shunted integratorStability condition:• unity gain freq. before second pole• otherwise feedback becomes positive

– oscillation

Real op amp

-

+

Vin Vout

log(Vout/Vin)

log(f )

Unity gain

Gain response

Phaseshift

log(f )Phase response

-90 degrees

0 degrees

Max gain

0

-180 degrees

Single pole6 dB

Double pole12 dB

Integrator with leadHigh frequency gain has minimum valuePurpose:• Provides phase lead• Can compensate for pole in servoAlternate circuits for lead• Capacitor at input, inductor in feedback• Overall positive phase

– analog to faster than light propagation– output anticipates input

Integrator with lead

Rin

-

+

Vin Vout

Cf

V1

Iin

If

Rf

log(Vout/Vin)

log(f )

Gain response

Phaseshift

log(f )Phase response

-90 degrees

0 degrees

Min gain

0

Single pole6 dB

Summing amplifier• High gain forces V1 ~ 0

• Feedback current must cancel input current

• Generalize to multiple inputs

– Vout = -Rfeedback Iin = -(Rf / Rin) Vin

– Works because V1 ~ 0

• Summing amplifier– Op amp input is summing junction

– Useful for combining multiple inputs

-

+

Rin1 Vin1

Vout

RfSummingjunction

If

Iin

Rin2 Vin2

Rin3 Vin3

Summing amplifier

V1

Op amp

-

+

Vout

Rf

inverting

non-inverting

If

IinV1

RinVin

Drift-compensated integratorReal op amps have leakage current• Can saturate integrator• Compensate with dc current to summing junction

Drift-compensated integrator

Rin

-

+

Vin

Vout

Cf

V1 Iin

If

Rc

Ic

+V

- V

Integratorleakagecompensation

Trans-impedance amplifiers• Input is current source

– model as voltage source with high impedance

– Iin = Vsource / Zsource

– Vout = -Zfeedback Vsource/ Zsource = -Iin Zfeedback

• Trans-impedance amplifier– current in, voltage out

– gain expressed in Ohms

V1

Trans-impedance amplifier

-

+

Vout

Rf

inverting

non-inverting

If

Iin

Currentsource V1

Iin

Vsource

Zsource

Photodiode amplifier• Photodiode like current source but with capacitor• Input capacitor causes op amp gain to diverge at high freq.

– Amplifies high freq noise– Oscillation

• Solution:– Shunt capacitor in feedback

Photo-diode amplifier

-

+

Vout

Rf

If

Ipd

Vbias

light

Photo-diodeequiv. circuit

Ipd

Cpd

Rpd

Shuntcapacitor

Cs

log(

Vou

t/Iin)

log(f )

Gain response

Unshunted

Shunted

Integrator design tipsShunting switch with small value resistor• Discharges capacitor

– initialize integrator/ servo– simplify servo testing allow rest of circuit

Resistors on power supply rails• Limits current to saturated op amp

– prevents burn outCapacitors on supply rails• Reduces noiseNote on capacitors• High values

– electrolytic, polar– leakage resistance

• Use low value in parallel

Shunt switch on integrator

Rin

-

+

Vin Vout

Cf

V1

Iin

If

Rs switch

+15V

-15V

Filtered and limited supply rails